# Properties

 Label 5472.2.a.bt.1.3 Level $5472$ Weight $2$ Character 5472.1 Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.32973$$ of defining polynomial Character $$\chi$$ $$=$$ 5472.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.844614 q^{5} +5.06562 q^{7} +O(q^{10})$$ $$q+0.844614 q^{5} +5.06562 q^{7} +2.84461 q^{11} +2.90210 q^{13} -7.72508 q^{17} -1.00000 q^{19} +3.91023 q^{23} -4.28663 q^{25} -2.90210 q^{29} +7.00814 q^{31} +4.27849 q^{35} +9.00814 q^{37} +6.81233 q^{41} -5.96772 q^{43} -1.04042 q^{47} +18.6605 q^{49} +9.03334 q^{53} +2.40260 q^{55} +0.749220 q^{59} +2.27849 q^{61} +2.45115 q^{65} -10.3739 q^{67} +2.13124 q^{71} +4.60197 q^{73} +14.4097 q^{77} -3.88503 q^{79} +8.44201 q^{83} -6.52470 q^{85} -16.0667 q^{89} +14.7009 q^{91} -0.844614 q^{95} -3.68923 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5} + q^{7}+O(q^{10})$$ 4 * q - q^5 + q^7 $$4 q - q^{5} + q^{7} + 7 q^{11} + 10 q^{13} - 5 q^{17} - 4 q^{19} - 8 q^{23} + 17 q^{25} - 10 q^{29} + 6 q^{31} + 5 q^{35} + 14 q^{37} + 2 q^{41} - 3 q^{43} - 3 q^{47} + 7 q^{49} - 4 q^{53} + 35 q^{55} + 20 q^{59} - 3 q^{61} + 12 q^{65} - 8 q^{67} - 30 q^{71} + 9 q^{73} + 7 q^{77} - 10 q^{79} + 4 q^{83} + 19 q^{85} + 16 q^{89} - 10 q^{91} + q^{95} - 6 q^{97}+O(q^{100})$$ 4 * q - q^5 + q^7 + 7 * q^11 + 10 * q^13 - 5 * q^17 - 4 * q^19 - 8 * q^23 + 17 * q^25 - 10 * q^29 + 6 * q^31 + 5 * q^35 + 14 * q^37 + 2 * q^41 - 3 * q^43 - 3 * q^47 + 7 * q^49 - 4 * q^53 + 35 * q^55 + 20 * q^59 - 3 * q^61 + 12 * q^65 - 8 * q^67 - 30 * q^71 + 9 * q^73 + 7 * q^77 - 10 * q^79 + 4 * q^83 + 19 * q^85 + 16 * q^89 - 10 * q^91 + q^95 - 6 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0.844614 0.377723 0.188861 0.982004i $$-0.439520\pi$$
0.188861 + 0.982004i $$0.439520\pi$$
$$6$$ 0 0
$$7$$ 5.06562 1.91462 0.957312 0.289056i $$-0.0933412\pi$$
0.957312 + 0.289056i $$0.0933412\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.84461 0.857683 0.428842 0.903380i $$-0.358922\pi$$
0.428842 + 0.903380i $$0.358922\pi$$
$$12$$ 0 0
$$13$$ 2.90210 0.804897 0.402449 0.915443i $$-0.368159\pi$$
0.402449 + 0.915443i $$0.368159\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −7.72508 −1.87361 −0.936803 0.349857i $$-0.886230\pi$$
−0.936803 + 0.349857i $$0.886230\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.91023 0.815340 0.407670 0.913129i $$-0.366341\pi$$
0.407670 + 0.913129i $$0.366341\pi$$
$$24$$ 0 0
$$25$$ −4.28663 −0.857326
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.90210 −0.538906 −0.269453 0.963014i $$-0.586843\pi$$
−0.269453 + 0.963014i $$0.586843\pi$$
$$30$$ 0 0
$$31$$ 7.00814 1.25870 0.629349 0.777123i $$-0.283322\pi$$
0.629349 + 0.777123i $$0.283322\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.27849 0.723197
$$36$$ 0 0
$$37$$ 9.00814 1.48093 0.740464 0.672096i $$-0.234606\pi$$
0.740464 + 0.672096i $$0.234606\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.81233 1.06391 0.531954 0.846773i $$-0.321458\pi$$
0.531954 + 0.846773i $$0.321458\pi$$
$$42$$ 0 0
$$43$$ −5.96772 −0.910069 −0.455034 0.890474i $$-0.650373\pi$$
−0.455034 + 0.890474i $$0.650373\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.04042 −0.151760 −0.0758802 0.997117i $$-0.524177\pi$$
−0.0758802 + 0.997117i $$0.524177\pi$$
$$48$$ 0 0
$$49$$ 18.6605 2.66579
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.03334 1.24082 0.620412 0.784276i $$-0.286965\pi$$
0.620412 + 0.784276i $$0.286965\pi$$
$$54$$ 0 0
$$55$$ 2.40260 0.323966
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0.749220 0.0975401 0.0487701 0.998810i $$-0.484470\pi$$
0.0487701 + 0.998810i $$0.484470\pi$$
$$60$$ 0 0
$$61$$ 2.27849 0.291731 0.145866 0.989304i $$-0.453403\pi$$
0.145866 + 0.989304i $$0.453403\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.45115 0.304028
$$66$$ 0 0
$$67$$ −10.3739 −1.26737 −0.633686 0.773590i $$-0.718459\pi$$
−0.633686 + 0.773590i $$0.718459\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.13124 0.252932 0.126466 0.991971i $$-0.459637\pi$$
0.126466 + 0.991971i $$0.459637\pi$$
$$72$$ 0 0
$$73$$ 4.60197 0.538620 0.269310 0.963054i $$-0.413204\pi$$
0.269310 + 0.963054i $$0.413204\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 14.4097 1.64214
$$78$$ 0 0
$$79$$ −3.88503 −0.437100 −0.218550 0.975826i $$-0.570133\pi$$
−0.218550 + 0.975826i $$0.570133\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 8.44201 0.926631 0.463316 0.886193i $$-0.346660\pi$$
0.463316 + 0.886193i $$0.346660\pi$$
$$84$$ 0 0
$$85$$ −6.52470 −0.707703
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −16.0667 −1.70306 −0.851532 0.524302i $$-0.824326\pi$$
−0.851532 + 0.524302i $$0.824326\pi$$
$$90$$ 0 0
$$91$$ 14.7009 1.54108
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −0.844614 −0.0866555
$$96$$ 0 0
$$97$$ −3.68923 −0.374584 −0.187292 0.982304i $$-0.559971\pi$$
−0.187292 + 0.982304i $$0.559971\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 0 0
$$103$$ 0.812333 0.0800415 0.0400208 0.999199i $$-0.487258\pi$$
0.0400208 + 0.999199i $$0.487258\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 9.88860 0.955967 0.477983 0.878369i $$-0.341368\pi$$
0.477983 + 0.878369i $$0.341368\pi$$
$$108$$ 0 0
$$109$$ −10.8375 −1.03805 −0.519024 0.854760i $$-0.673704\pi$$
−0.519024 + 0.854760i $$0.673704\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0.310773 0.0292351 0.0146175 0.999893i $$-0.495347\pi$$
0.0146175 + 0.999893i $$0.495347\pi$$
$$114$$ 0 0
$$115$$ 3.30264 0.307972
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −39.1323 −3.58725
$$120$$ 0 0
$$121$$ −2.90817 −0.264379
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −7.84361 −0.701554
$$126$$ 0 0
$$127$$ −18.6882 −1.65831 −0.829156 0.559017i $$-0.811178\pi$$
−0.829156 + 0.559017i $$0.811178\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.6055 −1.45083 −0.725416 0.688310i $$-0.758353\pi$$
−0.725416 + 0.688310i $$0.758353\pi$$
$$132$$ 0 0
$$133$$ −5.06562 −0.439245
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.91274 0.248852 0.124426 0.992229i $$-0.460291\pi$$
0.124426 + 0.992229i $$0.460291\pi$$
$$138$$ 0 0
$$139$$ 0.278492 0.0236214 0.0118107 0.999930i $$-0.496240\pi$$
0.0118107 + 0.999930i $$0.496240\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.25535 0.690347
$$144$$ 0 0
$$145$$ −2.45115 −0.203557
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 8.08269 0.662160 0.331080 0.943603i $$-0.392587\pi$$
0.331080 + 0.943603i $$0.392587\pi$$
$$150$$ 0 0
$$151$$ −15.0585 −1.22545 −0.612723 0.790297i $$-0.709926\pi$$
−0.612723 + 0.790297i $$0.709926\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 5.91917 0.475439
$$156$$ 0 0
$$157$$ 10.9273 0.872094 0.436047 0.899924i $$-0.356378\pi$$
0.436047 + 0.899924i $$0.356378\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 19.8078 1.56107
$$162$$ 0 0
$$163$$ −0.697363 −0.0546217 −0.0273108 0.999627i $$-0.508694\pi$$
−0.0273108 + 0.999627i $$0.508694\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.24621 0.483346 0.241673 0.970358i $$-0.422304\pi$$
0.241673 + 0.970358i $$0.422304\pi$$
$$168$$ 0 0
$$169$$ −4.57782 −0.352140
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.69736 −0.509191 −0.254596 0.967048i $$-0.581942\pi$$
−0.254596 + 0.967048i $$0.581942\pi$$
$$174$$ 0 0
$$175$$ −21.7144 −1.64146
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 18.2462 1.36379 0.681893 0.731452i $$-0.261157\pi$$
0.681893 + 0.731452i $$0.261157\pi$$
$$180$$ 0 0
$$181$$ 17.5651 1.30561 0.652803 0.757528i $$-0.273593\pi$$
0.652803 + 0.757528i $$0.273593\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 7.60839 0.559380
$$186$$ 0 0
$$187$$ −21.9749 −1.60696
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.19686 0.520747 0.260373 0.965508i $$-0.416154\pi$$
0.260373 + 0.965508i $$0.416154\pi$$
$$192$$ 0 0
$$193$$ −2.92730 −0.210712 −0.105356 0.994435i $$-0.533598\pi$$
−0.105356 + 0.994435i $$0.533598\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.1394 0.793648 0.396824 0.917895i $$-0.370112\pi$$
0.396824 + 0.917895i $$0.370112\pi$$
$$198$$ 0 0
$$199$$ −5.57220 −0.395003 −0.197501 0.980303i $$-0.563283\pi$$
−0.197501 + 0.980303i $$0.563283\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −14.7009 −1.03180
$$204$$ 0 0
$$205$$ 5.75379 0.401862
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.84461 −0.196766
$$210$$ 0 0
$$211$$ 1.23451 0.0849871 0.0424935 0.999097i $$-0.486470\pi$$
0.0424935 + 0.999097i $$0.486470\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.04042 −0.343754
$$216$$ 0 0
$$217$$ 35.5006 2.40993
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −22.4189 −1.50806
$$222$$ 0 0
$$223$$ 5.76092 0.385780 0.192890 0.981220i $$-0.438214\pi$$
0.192890 + 0.981220i $$0.438214\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −11.1862 −0.742455 −0.371228 0.928542i $$-0.621063\pi$$
−0.371228 + 0.928542i $$0.621063\pi$$
$$228$$ 0 0
$$229$$ 25.3624 1.67600 0.837999 0.545672i $$-0.183726\pi$$
0.837999 + 0.545672i $$0.183726\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.33804 −0.546243 −0.273122 0.961980i $$-0.588056\pi$$
−0.273122 + 0.961980i $$0.588056\pi$$
$$234$$ 0 0
$$235$$ −0.878750 −0.0573233
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.4441 −1.06368 −0.531839 0.846845i $$-0.678499\pi$$
−0.531839 + 0.846845i $$0.678499\pi$$
$$240$$ 0 0
$$241$$ −12.8286 −0.826363 −0.413182 0.910649i $$-0.635583\pi$$
−0.413182 + 0.910649i $$0.635583\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 15.7609 1.00693
$$246$$ 0 0
$$247$$ −2.90210 −0.184656
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 17.8527 1.12686 0.563428 0.826165i $$-0.309482\pi$$
0.563428 + 0.826165i $$0.309482\pi$$
$$252$$ 0 0
$$253$$ 11.1231 0.699304
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.49342 −0.0931572 −0.0465786 0.998915i $$-0.514832\pi$$
−0.0465786 + 0.998915i $$0.514832\pi$$
$$258$$ 0 0
$$259$$ 45.6318 2.83542
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0.483433 0.0298097 0.0149049 0.999889i $$-0.495255\pi$$
0.0149049 + 0.999889i $$0.495255\pi$$
$$264$$ 0 0
$$265$$ 7.62968 0.468688
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −8.45115 −0.515276 −0.257638 0.966242i $$-0.582944\pi$$
−0.257638 + 0.966242i $$0.582944\pi$$
$$270$$ 0 0
$$271$$ 16.5822 1.00730 0.503648 0.863909i $$-0.331991\pi$$
0.503648 + 0.863909i $$0.331991\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.1938 −0.735314
$$276$$ 0 0
$$277$$ 15.7124 0.944065 0.472032 0.881581i $$-0.343521\pi$$
0.472032 + 0.881581i $$0.343521\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.13938 −0.187280 −0.0936398 0.995606i $$-0.529850\pi$$
−0.0936398 + 0.995606i $$0.529850\pi$$
$$282$$ 0 0
$$283$$ 26.3884 1.56863 0.784315 0.620363i $$-0.213015\pi$$
0.784315 + 0.620363i $$0.213015\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 34.5087 2.03698
$$288$$ 0 0
$$289$$ 42.6768 2.51040
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −13.5903 −0.793955 −0.396978 0.917828i $$-0.629941\pi$$
−0.396978 + 0.917828i $$0.629941\pi$$
$$294$$ 0 0
$$295$$ 0.632801 0.0368431
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 11.3479 0.656265
$$300$$ 0 0
$$301$$ −30.2302 −1.74244
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1.92445 0.110193
$$306$$ 0 0
$$307$$ 3.44302 0.196503 0.0982516 0.995162i $$-0.468675\pi$$
0.0982516 + 0.995162i $$0.468675\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −29.4935 −1.67242 −0.836211 0.548408i $$-0.815234\pi$$
−0.836211 + 0.548408i $$0.815234\pi$$
$$312$$ 0 0
$$313$$ −10.0631 −0.568801 −0.284400 0.958706i $$-0.591794\pi$$
−0.284400 + 0.958706i $$0.591794\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4.14931 −0.233049 −0.116524 0.993188i $$-0.537175\pi$$
−0.116524 + 0.993188i $$0.537175\pi$$
$$318$$ 0 0
$$319$$ −8.25535 −0.462211
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 7.72508 0.429835
$$324$$ 0 0
$$325$$ −12.4402 −0.690059
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −5.27036 −0.290564
$$330$$ 0 0
$$331$$ −25.8886 −1.42297 −0.711483 0.702703i $$-0.751976\pi$$
−0.711483 + 0.702703i $$0.751976\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −8.76192 −0.478715
$$336$$ 0 0
$$337$$ 31.5097 1.71644 0.858221 0.513280i $$-0.171570\pi$$
0.858221 + 0.513280i $$0.171570\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 19.9354 1.07956
$$342$$ 0 0
$$343$$ 59.0677 3.18936
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −13.9173 −0.747120 −0.373560 0.927606i $$-0.621863\pi$$
−0.373560 + 0.927606i $$0.621863\pi$$
$$348$$ 0 0
$$349$$ 6.34305 0.339536 0.169768 0.985484i $$-0.445698\pi$$
0.169768 + 0.985484i $$0.445698\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 32.0794 1.70741 0.853707 0.520754i $$-0.174349\pi$$
0.853707 + 0.520754i $$0.174349\pi$$
$$354$$ 0 0
$$355$$ 1.80008 0.0955381
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.86670 −0.0985205 −0.0492603 0.998786i $$-0.515686\pi$$
−0.0492603 + 0.998786i $$0.515686\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.88689 0.203449
$$366$$ 0 0
$$367$$ −20.8840 −1.09014 −0.545069 0.838391i $$-0.683496\pi$$
−0.545069 + 0.838391i $$0.683496\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 45.7595 2.37571
$$372$$ 0 0
$$373$$ 25.9819 1.34529 0.672647 0.739964i $$-0.265157\pi$$
0.672647 + 0.739964i $$0.265157\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −8.42218 −0.433764
$$378$$ 0 0
$$379$$ −2.50301 −0.128571 −0.0642855 0.997932i $$-0.520477\pi$$
−0.0642855 + 0.997932i $$0.520477\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −21.8155 −1.11472 −0.557359 0.830272i $$-0.688185\pi$$
−0.557359 + 0.830272i $$0.688185\pi$$
$$384$$ 0 0
$$385$$ 12.1707 0.620274
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −35.2383 −1.78665 −0.893327 0.449407i $$-0.851635\pi$$
−0.893327 + 0.449407i $$0.851635\pi$$
$$390$$ 0 0
$$391$$ −30.2069 −1.52763
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3.28135 −0.165103
$$396$$ 0 0
$$397$$ −27.0263 −1.35641 −0.678205 0.734873i $$-0.737242\pi$$
−0.678205 + 0.734873i $$0.737242\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.8932 1.24311 0.621553 0.783372i $$-0.286502\pi$$
0.621553 + 0.783372i $$0.286502\pi$$
$$402$$ 0 0
$$403$$ 20.3383 1.01312
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 25.6247 1.27017
$$408$$ 0 0
$$409$$ −14.8336 −0.733475 −0.366738 0.930324i $$-0.619525\pi$$
−0.366738 + 0.930324i $$0.619525\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3.79526 0.186753
$$414$$ 0 0
$$415$$ 7.13024 0.350010
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6.05955 −0.296028 −0.148014 0.988985i $$-0.547288\pi$$
−0.148014 + 0.988985i $$0.547288\pi$$
$$420$$ 0 0
$$421$$ 13.8670 0.675834 0.337917 0.941176i $$-0.390278\pi$$
0.337917 + 0.941176i $$0.390278\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 33.1145 1.60629
$$426$$ 0 0
$$427$$ 11.5420 0.558555
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11.5580 −0.556729 −0.278364 0.960476i $$-0.589792\pi$$
−0.278364 + 0.960476i $$0.589792\pi$$
$$432$$ 0 0
$$433$$ −31.7468 −1.52565 −0.762826 0.646604i $$-0.776189\pi$$
−0.762826 + 0.646604i $$0.776189\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.91023 −0.187052
$$438$$ 0 0
$$439$$ 24.8840 1.18765 0.593825 0.804594i $$-0.297617\pi$$
0.593825 + 0.804594i $$0.297617\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −23.9911 −1.13985 −0.569926 0.821696i $$-0.693028\pi$$
−0.569926 + 0.821696i $$0.693028\pi$$
$$444$$ 0 0
$$445$$ −13.5701 −0.643286
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −32.1384 −1.51670 −0.758352 0.651845i $$-0.773995\pi$$
−0.758352 + 0.651845i $$0.773995\pi$$
$$450$$ 0 0
$$451$$ 19.3785 0.912496
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 12.4166 0.582099
$$456$$ 0 0
$$457$$ 29.1590 1.36400 0.681999 0.731353i $$-0.261111\pi$$
0.681999 + 0.731353i $$0.261111\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −18.6559 −0.868894 −0.434447 0.900697i $$-0.643056\pi$$
−0.434447 + 0.900697i $$0.643056\pi$$
$$462$$ 0 0
$$463$$ −22.6005 −1.05034 −0.525168 0.850999i $$-0.675997\pi$$
−0.525168 + 0.850999i $$0.675997\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −17.1666 −0.794377 −0.397189 0.917737i $$-0.630014\pi$$
−0.397189 + 0.917737i $$0.630014\pi$$
$$468$$ 0 0
$$469$$ −52.5502 −2.42654
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −16.9759 −0.780551
$$474$$ 0 0
$$475$$ 4.28663 0.196684
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −16.2625 −0.743052 −0.371526 0.928423i $$-0.621165\pi$$
−0.371526 + 0.928423i $$0.621165\pi$$
$$480$$ 0 0
$$481$$ 26.1425 1.19200
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −3.11597 −0.141489
$$486$$ 0 0
$$487$$ −18.2071 −0.825041 −0.412520 0.910948i $$-0.635351\pi$$
−0.412520 + 0.910948i $$0.635351\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 39.2493 1.77130 0.885649 0.464356i $$-0.153714\pi$$
0.885649 + 0.464356i $$0.153714\pi$$
$$492$$ 0 0
$$493$$ 22.4189 1.00970
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.7961 0.484270
$$498$$ 0 0
$$499$$ 35.7541 1.60057 0.800286 0.599619i $$-0.204681\pi$$
0.800286 + 0.599619i $$0.204681\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 27.5512 1.22845 0.614223 0.789132i $$-0.289470\pi$$
0.614223 + 0.789132i $$0.289470\pi$$
$$504$$ 0 0
$$505$$ 13.7218 0.610611
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 14.6328 0.648588 0.324294 0.945956i $$-0.394873\pi$$
0.324294 + 0.945956i $$0.394873\pi$$
$$510$$ 0 0
$$511$$ 23.3118 1.03125
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0.686107 0.0302335
$$516$$ 0 0
$$517$$ −2.95958 −0.130162
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.59926 0.332929 0.166465 0.986047i $$-0.446765\pi$$
0.166465 + 0.986047i $$0.446765\pi$$
$$522$$ 0 0
$$523$$ −19.8723 −0.868956 −0.434478 0.900682i $$-0.643067\pi$$
−0.434478 + 0.900682i $$0.643067\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −54.1384 −2.35830
$$528$$ 0 0
$$529$$ −7.71007 −0.335220
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 19.7701 0.856336
$$534$$ 0 0
$$535$$ 8.35204 0.361090
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 53.0820 2.28640
$$540$$ 0 0
$$541$$ 27.6982 1.19084 0.595420 0.803415i $$-0.296986\pi$$
0.595420 + 0.803415i $$0.296986\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −9.15353 −0.392094
$$546$$ 0 0
$$547$$ −33.8871 −1.44891 −0.724455 0.689322i $$-0.757908\pi$$
−0.724455 + 0.689322i $$0.757908\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.90210 0.123634
$$552$$ 0 0
$$553$$ −19.6801 −0.836883
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8.28763 0.351158 0.175579 0.984465i $$-0.443820\pi$$
0.175579 + 0.984465i $$0.443820\pi$$
$$558$$ 0 0
$$559$$ −17.3189 −0.732512
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −5.11397 −0.215528 −0.107764 0.994177i $$-0.534369\pi$$
−0.107764 + 0.994177i $$0.534369\pi$$
$$564$$ 0 0
$$565$$ 0.262483 0.0110427
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1.92830 −0.0808387 −0.0404194 0.999183i $$-0.512869\pi$$
−0.0404194 + 0.999183i $$0.512869\pi$$
$$570$$ 0 0
$$571$$ 23.3785 0.978358 0.489179 0.872183i $$-0.337297\pi$$
0.489179 + 0.872183i $$0.337297\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −16.7617 −0.699012
$$576$$ 0 0
$$577$$ −26.8807 −1.11906 −0.559530 0.828810i $$-0.689018\pi$$
−0.559530 + 0.828810i $$0.689018\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 42.7640 1.77415
$$582$$ 0 0
$$583$$ 25.6964 1.06423
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −36.7922 −1.51858 −0.759288 0.650754i $$-0.774453\pi$$
−0.759288 + 0.650754i $$0.774453\pi$$
$$588$$ 0 0
$$589$$ −7.00814 −0.288765
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 33.9034 1.39225 0.696123 0.717922i $$-0.254907\pi$$
0.696123 + 0.717922i $$0.254907\pi$$
$$594$$ 0 0
$$595$$ −33.0517 −1.35499
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −4.06456 −0.166073 −0.0830367 0.996546i $$-0.526462\pi$$
−0.0830367 + 0.996546i $$0.526462\pi$$
$$600$$ 0 0
$$601$$ −27.5560 −1.12403 −0.562016 0.827126i $$-0.689974\pi$$
−0.562016 + 0.827126i $$0.689974\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.45628 −0.0998621
$$606$$ 0 0
$$607$$ −25.5743 −1.03803 −0.519014 0.854766i $$-0.673701\pi$$
−0.519014 + 0.854766i $$0.673701\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.01939 −0.122152
$$612$$ 0 0
$$613$$ −31.2383 −1.26170 −0.630852 0.775903i $$-0.717295\pi$$
−0.630852 + 0.775903i $$0.717295\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.86189 0.235991 0.117995 0.993014i $$-0.462353\pi$$
0.117995 + 0.993014i $$0.462353\pi$$
$$618$$ 0 0
$$619$$ 31.6481 1.27204 0.636022 0.771671i $$-0.280579\pi$$
0.636022 + 0.771671i $$0.280579\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −81.3877 −3.26073
$$624$$ 0 0
$$625$$ 14.8083 0.592333
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −69.5885 −2.77468
$$630$$ 0 0
$$631$$ 3.86189 0.153739 0.0768696 0.997041i $$-0.475507\pi$$
0.0768696 + 0.997041i $$0.475507\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −15.7843 −0.626382
$$636$$ 0 0
$$637$$ 54.1546 2.14569
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14.9919 0.592143 0.296072 0.955166i $$-0.404323\pi$$
0.296072 + 0.955166i $$0.404323\pi$$
$$642$$ 0 0
$$643$$ 12.4906 0.492580 0.246290 0.969196i $$-0.420788\pi$$
0.246290 + 0.969196i $$0.420788\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23.5743 −0.926802 −0.463401 0.886149i $$-0.653371\pi$$
−0.463401 + 0.886149i $$0.653371\pi$$
$$648$$ 0 0
$$649$$ 2.13124 0.0836585
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.4693 −1.19236 −0.596178 0.802853i $$-0.703315\pi$$
−0.596178 + 0.802853i $$0.703315\pi$$
$$654$$ 0 0
$$655$$ −14.0253 −0.548012
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 39.9049 1.55447 0.777237 0.629209i $$-0.216621\pi$$
0.777237 + 0.629209i $$0.216621\pi$$
$$660$$ 0 0
$$661$$ −22.8156 −0.887422 −0.443711 0.896170i $$-0.646338\pi$$
−0.443711 + 0.896170i $$0.646338\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.27849 −0.165913
$$666$$ 0 0
$$667$$ −11.3479 −0.439392
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.48143 0.250213
$$672$$ 0 0
$$673$$ −37.0265 −1.42727 −0.713634 0.700519i $$-0.752952\pi$$
−0.713634 + 0.700519i $$0.752952\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 45.6733 1.75537 0.877683 0.479241i $$-0.159088\pi$$
0.877683 + 0.479241i $$0.159088\pi$$
$$678$$ 0 0
$$679$$ −18.6882 −0.717188
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 13.9837 0.535072 0.267536 0.963548i $$-0.413790\pi$$
0.267536 + 0.963548i $$0.413790\pi$$
$$684$$ 0 0
$$685$$ 2.46014 0.0939972
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 26.2156 0.998736
$$690$$ 0 0
$$691$$ 0.00185566 7.05926e−5 0 3.52963e−5 1.00000i $$-0.499989\pi$$
3.52963e−5 1.00000i $$0.499989\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0.235218 0.00892233
$$696$$ 0 0
$$697$$ −52.6258 −1.99334
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.0071 0.831198 0.415599 0.909548i $$-0.363572\pi$$
0.415599 + 0.909548i $$0.363572\pi$$
$$702$$ 0 0
$$703$$ −9.00814 −0.339748
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 82.2971 3.09510
$$708$$ 0 0
$$709$$ 3.94045 0.147987 0.0739934 0.997259i $$-0.476426\pi$$
0.0739934 + 0.997259i $$0.476426\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 27.4035 1.02627
$$714$$ 0 0
$$715$$ 6.97258 0.260760
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −29.0656 −1.08396 −0.541982 0.840390i $$-0.682326\pi$$
−0.541982 + 0.840390i $$0.682326\pi$$
$$720$$ 0 0
$$721$$ 4.11497 0.153249
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 12.4402 0.462018
$$726$$ 0 0
$$727$$ −10.1617 −0.376877 −0.188439 0.982085i $$-0.560343\pi$$
−0.188439 + 0.982085i $$0.560343\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 46.1011 1.70511
$$732$$ 0 0
$$733$$ 51.4777 1.90137 0.950686 0.310156i $$-0.100381\pi$$
0.950686 + 0.310156i $$0.100381\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −29.5097 −1.08700
$$738$$ 0 0
$$739$$ −33.1412 −1.21912 −0.609560 0.792740i $$-0.708654\pi$$
−0.609560 + 0.792740i $$0.708654\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 35.5006 1.30239 0.651195 0.758911i $$-0.274268\pi$$
0.651195 + 0.758911i $$0.274268\pi$$
$$744$$ 0 0
$$745$$ 6.82675 0.250113
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 50.0919 1.83032
$$750$$ 0 0
$$751$$ −15.2594 −0.556822 −0.278411 0.960462i $$-0.589808\pi$$
−0.278411 + 0.960462i $$0.589808\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.7187 −0.462879
$$756$$ 0 0
$$757$$ 7.28161 0.264655 0.132327 0.991206i $$-0.457755\pi$$
0.132327 + 0.991206i $$0.457755\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.47886 −0.343609 −0.171804 0.985131i $$-0.554960\pi$$
−0.171804 + 0.985131i $$0.554960\pi$$
$$762$$ 0 0
$$763$$ −54.8989 −1.98747
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2.17431 0.0785098
$$768$$ 0 0
$$769$$ 19.2760 0.695112 0.347556 0.937659i $$-0.387012\pi$$
0.347556 + 0.937659i $$0.387012\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −9.19501 −0.330721 −0.165361 0.986233i $$-0.552879\pi$$
−0.165361 + 0.986233i $$0.552879\pi$$
$$774$$ 0 0
$$775$$ −30.0413 −1.07911
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.81233 −0.244077
$$780$$ 0 0
$$781$$ 6.06256 0.216935
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9.22935 0.329410
$$786$$ 0 0
$$787$$ −12.6847 −0.452159 −0.226080 0.974109i $$-0.572591\pi$$
−0.226080 + 0.974109i $$0.572591\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.57426 0.0559741
$$792$$ 0 0
$$793$$ 6.61241 0.234814
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −27.4591 −0.972651 −0.486325 0.873778i $$-0.661663\pi$$
−0.486325 + 0.873778i $$0.661663\pi$$
$$798$$ 0 0
$$799$$ 8.03730 0.284339
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 13.0908 0.461965
$$804$$ 0 0
$$805$$ 16.7299 0.589652
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −18.2912 −0.643084 −0.321542 0.946895i $$-0.604201\pi$$
−0.321542 + 0.946895i $$0.604201\pi$$
$$810$$ 0 0
$$811$$ −6.28391 −0.220658 −0.110329 0.993895i $$-0.535190\pi$$
−0.110329 + 0.993895i $$0.535190\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −0.589002 −0.0206318
$$816$$ 0 0
$$817$$ 5.96772 0.208784
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 43.9103 1.53248 0.766240 0.642555i $$-0.222125\pi$$
0.766240 + 0.642555i $$0.222125\pi$$
$$822$$ 0 0
$$823$$ 26.2483 0.914957 0.457479 0.889221i $$-0.348753\pi$$
0.457479 + 0.889221i $$0.348753\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −34.7726 −1.20916 −0.604581 0.796543i $$-0.706660\pi$$
−0.604581 + 0.796543i $$0.706660\pi$$
$$828$$ 0 0
$$829$$ −20.9184 −0.726525 −0.363263 0.931687i $$-0.618337\pi$$
−0.363263 + 0.931687i $$0.618337\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −144.154 −4.99464
$$834$$ 0 0
$$835$$ 5.27563 0.182571
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 41.0061 1.41569 0.707844 0.706368i $$-0.249668\pi$$
0.707844 + 0.706368i $$0.249668\pi$$
$$840$$ 0 0
$$841$$ −20.5778 −0.709580
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3.86649 −0.133011
$$846$$ 0 0
$$847$$ −14.7317 −0.506187
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 35.2239 1.20746
$$852$$ 0 0
$$853$$ 11.5056 0.393943 0.196972 0.980409i $$-0.436889\pi$$
0.196972 + 0.980409i $$0.436889\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −14.7478 −0.503774 −0.251887 0.967757i $$-0.581051\pi$$
−0.251887 + 0.967757i $$0.581051\pi$$
$$858$$ 0 0
$$859$$ −36.3402 −1.23991 −0.619955 0.784637i $$-0.712849\pi$$
−0.619955 + 0.784637i $$0.712849\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 48.3454 1.64570 0.822849 0.568260i $$-0.192383\pi$$
0.822849 + 0.568260i $$0.192383\pi$$
$$864$$ 0 0
$$865$$ −5.65668 −0.192333
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −11.0514 −0.374893
$$870$$ 0 0
$$871$$ −30.1060 −1.02010
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −39.7328 −1.34321
$$876$$ 0 0
$$877$$ −4.23226 −0.142913 −0.0714567 0.997444i $$-0.522765\pi$$
−0.0714567 + 0.997444i $$0.522765\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.6651 −1.10051 −0.550257 0.834995i $$-0.685470\pi$$
−0.550257 + 0.834995i $$0.685470\pi$$
$$882$$ 0 0
$$883$$ 45.3320 1.52554 0.762772 0.646668i $$-0.223838\pi$$
0.762772 + 0.646668i $$0.223838\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −50.5271 −1.69653 −0.848267 0.529569i $$-0.822354\pi$$
−0.848267 + 0.529569i $$0.822354\pi$$
$$888$$ 0 0
$$889$$ −94.6675 −3.17504
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.04042 0.0348162
$$894$$ 0 0
$$895$$ 15.4110 0.515133
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −20.3383 −0.678320
$$900$$ 0 0
$$901$$ −69.7832 −2.32482
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 14.8357 0.493157
$$906$$ 0 0
$$907$$ 21.9823 0.729910 0.364955 0.931025i $$-0.381084\pi$$
0.364955 + 0.931025i $$0.381084\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −29.9950 −0.993778 −0.496889 0.867814i $$-0.665524\pi$$
−0.496889 + 0.867814i $$0.665524\pi$$
$$912$$ 0 0
$$913$$ 24.0143 0.794756
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −84.1174 −2.77780
$$918$$ 0 0
$$919$$ 30.5481 1.00769 0.503844 0.863795i $$-0.331919\pi$$
0.503844 + 0.863795i $$0.331919\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 6.18507 0.203584
$$924$$ 0 0
$$925$$ −38.6145 −1.26964
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 5.67151 0.186076 0.0930380 0.995663i $$-0.470342\pi$$
0.0930380 + 0.995663i $$0.470342\pi$$
$$930$$ 0 0
$$931$$ −18.6605 −0.611574
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −18.5603 −0.606985
$$936$$ 0 0
$$937$$ 33.6051 1.09783 0.548915 0.835878i $$-0.315041\pi$$
0.548915 + 0.835878i $$0.315041\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −30.0543 −0.979743 −0.489871 0.871795i $$-0.662956\pi$$
−0.489871 + 0.871795i $$0.662956\pi$$
$$942$$ 0 0
$$943$$ 26.6378 0.867447
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.3058 −0.399883 −0.199942 0.979808i $$-0.564075\pi$$
−0.199942 + 0.979808i $$0.564075\pi$$
$$948$$ 0 0
$$949$$ 13.3554 0.433534
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −30.4674 −0.986937 −0.493468 0.869764i $$-0.664271\pi$$
−0.493468 + 0.869764i $$0.664271\pi$$
$$954$$ 0 0
$$955$$ 6.07857 0.196698
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 14.7548 0.476459
$$960$$ 0 0
$$961$$ 18.1140 0.584322
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2.47244 −0.0795906
$$966$$ 0 0
$$967$$ 5.26560 0.169330 0.0846652 0.996409i $$-0.473018\pi$$
0.0846652 + 0.996409i $$0.473018\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −54.7895 −1.75828 −0.879139 0.476566i $$-0.841881\pi$$
−0.879139 + 0.476566i $$0.841881\pi$$
$$972$$ 0 0
$$973$$ 1.41074 0.0452261
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −28.9690 −0.926800 −0.463400 0.886149i $$-0.653371\pi$$
−0.463400 + 0.886149i $$0.653371\pi$$
$$978$$ 0 0
$$979$$ −45.7035 −1.46069
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 35.2997 1.12589 0.562943 0.826495i $$-0.309669\pi$$
0.562943 + 0.826495i $$0.309669\pi$$
$$984$$ 0 0
$$985$$ 9.40847 0.299779
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −23.3352 −0.742016
$$990$$ 0 0
$$991$$ 8.34833 0.265194 0.132597 0.991170i $$-0.457668\pi$$
0.132597 + 0.991170i $$0.457668\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.70635 −0.149201
$$996$$ 0 0
$$997$$ 5.81519 0.184169 0.0920845 0.995751i $$-0.470647\pi$$
0.0920845 + 0.995751i $$0.470647\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5472.2.a.bt.1.3 4
3.2 odd 2 608.2.a.i.1.3 4
4.3 odd 2 5472.2.a.bs.1.3 4
12.11 even 2 608.2.a.j.1.1 yes 4
24.5 odd 2 1216.2.a.x.1.2 4
24.11 even 2 1216.2.a.w.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.3 4 3.2 odd 2
608.2.a.j.1.1 yes 4 12.11 even 2
1216.2.a.w.1.4 4 24.11 even 2
1216.2.a.x.1.2 4 24.5 odd 2
5472.2.a.bs.1.3 4 4.3 odd 2
5472.2.a.bt.1.3 4 1.1 even 1 trivial